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The rotation is completely specified by specifying the axis planes and the angles of rotation about them. Without loss of generality, these axis planes may be chosen to be the uz - and xy-planes of a Cartesian coordinate system, allowing a simpler visualization of the rotation. In 4D space, the Hopf angles {ξ 1, η, ξ 2} parameterize the 3 ...
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
Einstein's concept of spacetime has a Minkowski structure based on a non-Euclidean geometry with three spatial dimensions and one temporal dimension, rather than the four symmetric spatial dimensions of Schläfli's Euclidean 4D space. Single locations in Euclidean 4D space can be given as vectors or 4-tuples, i.e., as ordered lists of numbers ...
1951, A. C. Hurley, Finite rotation groups and crystal classes in four dimensions, Proceedings of the Cambridge Philosophical Society, vol. 47, issue 04, p. 650 [1] 1962 A. L. MacKay Bravais Lattices in Four-dimensional Space [2] 1964 Patrick du Val, Homographies, quaternions and rotations, quaternion-based 4D point groups
A rotation can be represented by a unit-length quaternion q = (w, r →) with scalar (real) part w and vector (imaginary) part r →. The rotation can be applied to a 3D vector v → via the formula = + (+). This requires only 15 multiplications and 15 additions to evaluate (or 18 multiplications and 12 additions if the factor of 2 is done via ...
A plane rotation around a point followed by another rotation around a different point results in a total motion which is either a rotation (as in this picture), or a translation. A motion of a Euclidean space is the same as its isometry : it leaves the distance between any two points unchanged after the transformation.
The two rotation planes span four-dimensional space, so every point in the space can be specified by two points, one on each of the planes. A double rotation has two angles of rotation, one for each plane of rotation. The rotation is specified by giving the two planes and two non-zero angles, α and β (if either angle is zero the rotation is ...
Poloidal direction (red arrow) and toroidal direction (blue arrow) A torus of revolution in 3-space can be parametrized as: [2] (,) = (+ ) (,) = (+ ) (,) = using angular coordinates θ, φ ∈ [0, 2π), representing rotation around the tube and rotation around the torus's axis of revolution, respectively, where the major radius R is the distance from the center of the tube to ...